Counting Add Summation

1. Counting Add Summation An Introduction

2. Counting • Here counting doesn’t mean counting the things physically. • Here we will learn, how we can count without counting. • Two basic things regarding counting are: • Sequences • Series

3. Sequence • Let N be the set of natural numbers and Nn be the set of first n natural numbers, i.e., Nn ={1,2,3,4,……….n} and X be a non-empty set, then a map f:NnX is called a finite sequence and a map f:N X is called an infinite sequence. • A sequence following some definite rule (or rules) is called a “progression”.

4. Sequence • Illustrations: • 3,5,7,9,……………….,21 • 8,5,2,-1,-4,………..,16 • 2,6,18,54,………,1458 • 1,4,9,16,……………… • 1,3,5,7,9,…………….. • 8,5,2,-1,-4,……….. }Finite sequence }Infinite sequence

5. Series • If the terms of sequence are connected by plus (or minus) signs, a series is formed. • Thus, if Tn denotes the general term of a sequence, then T1 + T2 + T3 + T4 +………+ Tn is a series of n terms.

6. Series • Illustrations: • 3+5+7+9+…………+21 • 8+5+2+1+4+………+16 • 2+6+18+54+………+1458 • 1+4+9+16+…………. • 1+3+5+7+9+………... }Finite Series }Infinite Series

7. Arithmetic Progression • Arithmetic Progression(A.P.): A sequence (finite or infinite) is called an arithmetic progression ( abbreviated A.P.). Iff the difference of any term from its preceding term is constant. • This constant is usually denoted by d and is called common difference. • The first term of A.P. is usually denoted by a. • for example: The sequence 3,5,7,9,…..21 is a finite A.P. with d=2 • and the sequence 8,5,2,-1,…….. Is an infinite A.P. with d= -3

8. General terms of A.P. • Let a be the first term and d the common difference of an A.P. let T1 , T2 , T3 ,…………. Tn denote 1st , 2nd ,3rd ,……. nthterms respectively, then we have. T2 – T1 = d T3 – T2 = d T4 – T3 = d ……………. ……………. Tn – Tn-1 = d • General Term= Tn = a+(n-1)d

9. General terms of A.P. • If l is the last term of A.P., then the number of terms in the A.P. is • n= l-a+d /d • And the common difference d= l-a/n-1 • If a1 , a2 , a3 ,……….. ,an are non-zero numbers such that 1/ a1 ,1/ a2 ,1/ a3 ,…………, 1/ an are in A.P., then a1 , a2 , a3 ,……… ,an are said to be in “Harmonic Progression” (abbreviated H.P.) l-a d=--------- n-1 l-a+d n= -------- d

10. Sum of n terms of an A.P. • let a be the first term, d the common difference and l the last term of the given A.P. then Sn , sum of n terms of this A.P. can be calculated by three different mechanism depending upon input values : • If a, n and d is given • If a, n and l is given • If n, l and d is given Sn = n/2{2l-(n-1)d}

11. Arithmetic Means A.M. • When three numbers are in A.P., the middle one is said to be the Arithmetic Mean between the other two. • if a,b and c are in A.P., then A.M.= b • To find the A.M. between two given numbers A= a+b/2 where a and b are two given numbers. • if we have n terms between a and b then • A= n{(a+b)/2}

12. Geometric Progression G.P. • A sequence (finite or infinite) of non-zero terms is called a “geometric progression” iff the ratio of any terms to its preceding term is constant. • this (non-zero) constant is usually denoted by r and is called “Common ration” • We assume that none of the terms of the sequence is zero. • Eg. a,ar,ar2 , ar3 , ar4 ,…………., arn-1 is in G.P.

13. General terms of G.P. • Let a be the first term and r bethe common ratio of a G.P. let T1 , T2 , T3 ,…………. Tn denote 1st , 2nd ,3rd ,……. nthterms respectively, then we have. T2 = T1r T3= T2r T4 =T3r ……………. ……………. Tn=Tn-1r • General Term= Tn = arn-1

14. Sum of n terms of a G.P. • Sum of n terms of a G.P. depends upon the value of r. there are three possibilities as: • If r <1 • If r ≤ -1 or r >1 • If r=1

15. Summation